There are many different known methods for optical testing of samples. Interferometry is one of the better known methods.
In interferometric surface testing, a lightwave reflected by the surface of an object and a lightwave reflected by a reference surface are superimposed in an interferometric manner. The resulting pattern of varying light intensities contains information as to the deviations of the object surface from the reference surface. This brightness pattern is usually recorded by a camera.
Interferometric testing may similarly be carried out by using transmitted light. A light beam is split into a measuring beam portion and a reference beam portion, and the measuring beam is passed through a transparent sample. Then both optical paths are superimposed in an interferometric manner to generate a brightness pattern which is recorded by a camera.
The image of the brightness pattern is used to calculate phase values--associated with the brightness of each point (pixel) of the camera image--which are combined to form a phase image. Various methods are known for the calculation of these phase values. Some of these methods, as well as their advantages and disadvantages, have been discussed in a well-known thesis by B. Dorband, University of Stuttgart (1986).
Due to the periodicity of the interference equation, the phase value belonging to a particular level of brightness can, however, only be calculated up to an integral multiple of the number 2.pi., i.e. in modulo 2.pi.. If this unknown integral multiple is set equal to zero, even the phase images for objects whose surfaces are continuous will exhibit so-called "discontinuities". At these discontinuities, the difference between the calculated phase values of adjacent points has an absolute value greater than the number .pi.. To generate a phase map of the contours of the sample surface or of the deviations of the sample from a reference element, the proper integer multiple of 2.pi. must be determined for the phase values, i.e., the elimination of discontinuities is necessary.
In Applied Optics, Vol. 21, No. 14, Page 2470 (1982), K. Itoh has described a method for the elimination of discontinuities in a measuring system in which the camera records the image of the pattern along only one line: initially only the differences between the phase values of adjacent points of the camera image are calculated. Based on the sampling theorem, these differences must have an absolute value smaller than .pi. in order to be able to identify unambiguously those discontinuities in which the difference has an absolute value greater than .pi.. Therefore, where such discontinuities occur, the number 2.pi. is added to or subtracted from these differences so that the corrected differences between the phase values are between -.pi. and +.pi.. In this manner, these differences are expressed in terms of modulo 2.pi.. By integrating these phase difference values over the entire phase image, a phase map without discontinuities is ultimately obtained. In such a phase map, the integral multiple of the number 2.pi. of the phase values is determined, and the deviations of the object surface from the reference surface can be calculated in an unambiguous manner.
An extension of this Itoh method to two-dimensional camera images has been described by D. C. Ghiglia et al., Journal of the Optical Society of America, Vol. 4, No. 1, Page 267 (1987). As explained in that paper, the differences between the phase values of adjacent data points in lines, as well as in columns, are calculated; and these differences are expressed in modulo 2.pi.. A final phase map is obtained by path integration via the differences between the phase values or by iterative integration by means of cellular automatic systems; the latter process, however, is considerably slower than path integration.
If the output signals of the camera are noisy, inconsistencies, i.e., wrongly identified discontinuities, may occur. The difference between the phase values of adjacent data points may, for example, have an absolute value greater than .pi., even though there is no discontinuity. If the phase-value integration includes these inconsistencies, the calculation of integral-multiple phase differences would depend upon the particular path of integration. That is, the use of a few data points exhibiting inconsistencies caused by noisy measured values can result in a global corruption of the final phase map.
In order to suppress part of the inconsistent measured values, the authors (Ghiglia et al.) calculate the sign-correct sum of the modulo 2.pi. differences between the phase values along a closed path around partial fields consisting of each set of four adjacent data points throughout the entire data field. If the sum does not equal zero, all four data points are masked and are no longer considered during further evaluation. In mathematical terms, the vector field of the modulo 2.pi. differences between the phase values must not exhibit local vortexes. The inconsistent measured values for the masked data points are not considered in the phase-value integration. This prevents global corruption of the phase map by these inconsistent measured values. However, this method identifies only a part of the inconsistent measured values; and, while these are omitted from the final integration, the authors of this paper have conceded that global corruption of the phase map may still result from the remaining inconsistent measured values.
Numerous other publications, among others German Patent No. DE OS 36 00 672 and European Patent No. EP OS 0 262 089, have disclosed the projection of a bar pattern on the surface of an object and the recording of said bar pattern by a camera. The contours of the object surface cause deformations of the bar pattern recorded by the camera, and the evaluation of the camera image is analogous to the evaluation of interferometrically generated brightness patterns. Namely, a phase value of the bar pattern is first calculated from the brightness of each point (pixel) of the camera image, and then the calculated phase values are combined to form a phase image. This phase image also has discontinuities, because the phase values can be calculated only up to an integer multiple of the number 2.pi.; and in these methods the discontinuities can also be identified erroneously when the measured phase values are noisy.